Problem: Let N be a positive odd number.

There are N coins, numbered 1, 2, \ldots, N. For each i (1 \leq i \leq N), when Coin i is tossed, it comes up heads with probability p_i and tails with probability 1 - p_i.

Taro has tossed all the N coins. Find the probability of having more heads than tails.

Constraints

* N is an odd number.
* 1 \leq N \leq 2999
* p_i is a real number and has two decimal places.
* 0 < p_i < 1

Input

Input is given from Standard Input in the following format:


N
p_1 p_2 \ldots p_N


Output

Print the probability of having more heads than tails. The output is considered correct when the absolute error is not greater than 10^{-9}.

Examples

Input

3
0.30 0.60 0.80


Output

0.612


Input

1
0.50


Output

0.5


Input

5
0.42 0.01 0.42 0.99 0.42


Output

0.3821815872

Solution (in CPP):

Solution:
#include<bits/stdc++.h>
using namespace std;

const int N=100005;
const int MOD=1e9+7;
int n;
double a[3005],dp[3005][3005],ans;

int main()
{
	scanf("%d",&n);
	for(int i=1;i<=n;i++) scanf("%lf",&a[i]);
	dp[0][0]=1;
	for(int i=1;i<=n;i++)
	{
		dp[i][0]=dp[i-1][0]*(1-a[i]);
		for(int j=1;j<=i;j++)
		{
			dp[i][j]=dp[i-1][j]*(1-a[i])+a[i]*dp[i-1][j-1];
		}
	}
	for(int i=0;i<=(n-1)/2;i++) ans+=dp[n][n-i];
	printf("%.12lf\n",ans);
}